Minimax properties of Dirichlet kernel density estimators

This paper considers the asymptotic behavior in /3-Holder spaces, and under Lp losses, of a Dirichlet kernel density estimator proposed by Aitchison and Lauder (1985) for the analysis of compositional data. In recent work, Ouimet and Tolosana-Delgado (2022) established the uniform strong consistency and asymptotic normality of this estimator. As a complement, it is shown here that the Aitchison-Lauder estimator can achieve the minimax rate asymptotically for a suitable choice of bandwidth whenever (p, /3) E [1, 3) x (0, 2] or (p, /3) E Ad, where Ad is a specific subset of [3, 4) x (0, 2] that depends on the dimension d of the Dirichlet kernel. It is also shown that this estimator cannot be minimax when either p E [4, infinity) or /3 E (2, infinity). These results extend to the multivariate case, and also rectify in a minor way, earlier findings of Bertin and Klutchnikoff (2011) concerning the minimax properties of Beta kernel estimators. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).